The paper deals with the dual problem of a pair of rods made of a linear elastic and homogeneous material in natural vibrations. That is to answer what kind of crosssectional variation and homogeneous boundaries the two rods should have so that they have the same natural frequencies. Based on the dual of displacement description and internal force description, the paper presents the crosssectional variation and homogeneous boundaries for a dual of rods with different crosssections and the classification of all rods, including the dual of a fixedfixed rod and a freefree rod and the dual of a fixedfree rod and a freefixed rod. The two rods in a dual have the same natural frequencies while their displacement mode shapes are the position derivatives of each other. Then, the paper gives the formula of crosssectional area for a dual of rods with identical crosssections. In such a case, a fixedfixed rod and a freefree rod are a dual while a fixedfree rod and a freefixed rod are a pair of mirrors. The rods with uniform crosssections have the above dual properties by nature. Finally, the paper extends the above studies to the dual problem of a pair of arbitrary rods with both crosssection and material properties varying along their axes. The conclusions in the paper hold also true for the duality relations of circular shafts with homogeneous boundaries in natural vibrations.
参考文献
相似文献
引证文献
引用本文
胡海岩.杆在固有振动中的对偶关系[J].动力学与控制学报,2020,18(2):1~8; Hu Haiyan. Duality relations of rods in natural vibrations[J]. Journal of Dynamics and Control,2020,18(2):1-8.